The most important application of the Copernican anthropic principle is the
so-called doomsday argument (DA). It has been independently discovered at least three
times. Brandon Carter was first, but he did not publish. The other independent
co-discoverers are H. B. Nielsen and Richard Gott.

The credit for being the first person to clearly enunciate it in print belongs to John
Leslie who had heard about Carter's discovery from Frank Tipler. Leslie has been by far
the most prolific writer on the topic with one monograph and a dozen or so academic
papers.

Nielsen has only hinted at the DA in print [1989]. Gott has published a couple of
articles specifically about the DA (Gott [1993, 1997]). As we shall see, there are some
differences in how Gott and Leslie present the DA. Its clear that they put forth the
same argument, but they approach the issue from somewhat different angles.

The basic idea behind the DA is easy enough to grasp:

Imagine that two big urns are put in front of you, and you know that one of them
contains ten balls and the other a million, but you are ignorant as to which is which. You
know the balls in each urn are numbered 1, 2, 3, 4 ... etc. Now you take a ball at random
from the left urn, and it is number 7. Clearly, this is a strong indication that that urn
contains only ten balls. If originally the odds were fifty-fifty, a swift application of
Bayes' theorem gives you the posterior probability that the left urn is the one with only
ten balls. (pposterior (L=10) = 0.999990). But now consider the case where
instead of the urns you have two possible human races, and instead of balls you have
individuals, ranked according to birth order. As a matter of fact, you happen to find that
your rank is about sixty billion. Now, say Carter and Leslie, we should reason in the same
way as we did with the urns. That you should have a rank of sixty billion or so is much
more likely if only 100 billion persons will ever have lived than if there will be many
trillion persons. Therefore, by Bayes' theorem, you should update your beliefs about
humankinds prospects and realize that an impending doomsday is much more probable
than you have hitherto thought.

While the core idea can thus easily be stated in a single paragraph, a large part of
what makes up the corpus of "the argument" consists of replies to numerous
objections. If we regard these replies as part of the argument then a concise exposition
would easily fill a chapter of a book, even without any attempt to evaluate the various
claims that have been made.

Here I will only mention those objections that seem most alive. The ones that arguably
haven't been uncontroversially refuted. For a more exhaustive list of objections, see
Leslie [1996], chapters 5 and 6).

My exposition will take following path: We begin by looking at Gotts version of
the argument. We then move on to Leslies version and well see how he supports
it by imaginative analogies and thought experiments. Then we consider the most important
objections that have been advanced  by William Eckhardt, Dennis Dieks, Korb &
Oliver and others. We shall see how the proponents of the DA have answered these
objections. One set of objections, the ones related to the so-called Shooting-room paradox
(which Leslie attributes to Derek Parfit) is given a separate section at the end since it
introduces a new thought experiment and number of new issues.

The doomsday argument as presented by Richard Gott III

Astrophysicist Richard Gott III, who independently discovered the DA, first published
his ideas in a brilliant Nature paper [1993] (see also the responses and
Gotts replies: Goodman [1994], Buch [1994], Mackay [1994], Gott [1994]) and later
popularized some of them in an article in New Scientist [1997]. In the Nature paper
he not only sets forth a version of the DA but he also considers its implications for the
search of extraterrestrial life (SETI) and for the prospects of space travel. Here we will
focus on what he has to say about the DA. Gotts version of the DA is based on a more
general argument type that he calls the delta t argument.

The Delta t argument

Gott first explains an argument form that he calls the "Delta t
argument". It is extremely simple and yet Gott thinks it can be applied to make a
very wide range of predictions about most everything in heaven and on earth. It goes as
follows:

Suppose we want to estimate how long some series of observations (measurements) is
going to last. Then,

Assuming that whatever we are measuring can be observed only in the interval between
times tbegin and tend, if there is nothing special about tnow
we expect tnow to be randomly located in this interval. (p. 315)

Using this randomness assumption, we can make the estimate tfuture = (tend
 tnow) = tpast = (tnow  tbegin). tfuture
is the estimated value of how much longer the series will last. This means that we make
the estimate that the series will continue for as long as it has already lasted when we
make the random observation. This estimate will overestimate the true value half the time
and underestimate it half the time. It also follows that a 50% confidence interval is
given by

1/3 tpast < tfuture < 3 tpast

And a 95% confidence interval is given by

1/39 tpast < tfuture < 39 tpast

Gott gives some illustrations of how this reasoning can be applied in the real
world:

[In] 1969 I saw for the first time Stonehenge (tpast » 3,868 years) and the
Berlin Wall (tpast = 8 years). Assuming that I am a random observer of the
Wall, I expect to be located randomly in the time between tbegin and tend
(tend occurs when the Wall is destroyed or there are no visitors left to
observe it, whichever comes first). (p. 315)

At least in these two cases, the delta t argument seems to have worked! The New
Scientist article also features an inset that invites the reader to use the arrival
date of that issue of the magazine to predict how long their current relationship will
last. You can presumably use my paper for the same purpose. How long has your present
relationship lasted? Use that value for tpast and you get your prediction from
the expressions above, with the precise confidence intervals.

Wacky? Yes, but all this does indeed follow from the assumption that tnow is
randomly sampled from the interval tbegin to tend. This imposes some
restrictions on the applicability of the delta t argument:

[At] a friends wedding, you couldnt use the formula to forecast the
marriages future. You are at the wedding precisely to witness its beginning. Neither
can you use it to predict the future of the Universe itself  for intelligent
observers emerged only long after the Big Bang, and so witness only a subset of its
timeline. (Gott [1997], p. 39)

Gott does not discuss in any more detail the all-important question of when, in
practice, the delta t argument is applicable. We shall return to this issue in a
later chapter.

The Copernican anthropic principle

Underlying the delta t argument is what Gott calls the Copernican anthropic
principle, which says that you should consider yourself as being randomly sampled from
the set of all intelligent observers:

[T]he location of your birth in space and time in the Universe is privileged (or
special) only to the extent implied by the fact that you are an intelligent observer, that
your location among intelligent observers is not special but rather picked at random from
the set of all intelligent observers (past, present and future any one of whom you could
have been. (p. 316)

The Copernican anthropic principle says that you are more likely to be where there are
many observers that where there are few. This can be seen as a strengthening of the weak
anthropic principle, which says that you will be where there are observers. The Copernican
anthropic principle and the weak anthropic principle both assert that the prior
probability that you should be find yourself as anything other than an observer is zero.
But whereas the weak anthropic principle is silent as to the prior probability that you
should find yourself as a particular observer, the Copernican anthropic principle makes an
assertion about this too. It says that this prior probability should be 1/N, where N is
the total number of observers that will ever have existed. In other words, the Copernican
anthropic principle says that all (intelligent) observers should be assigned equal
sample density.

The doomsday argument as presented by Gott

If we want to apply the delta t argument to the life-expectancy of the human
species, we have to measure time on a "population clock" where one unit of time
corresponds to the birth of one human. This modification is necessary because the human
population is not constant. Due to population growth, most humans that have been born so
far find themselves later rather than earlier in the history of our species. According to
the Copernican anthropic principle, we should consequently assign a higher prior
probability to finding ourselves at these later times. By measuring time as the number of
births, we regain a scale where you should assign a uniform sampling density to all points
of time.

There have been something like 70 billion humans so far. Using this value as tpast,
the delta t argument gives the 95% confidence interval

billion < tfuture < 2.7 trillion.

The units here are human beings. In order to convert this to years, we would have to
estimate what the future population figures will be given that a total of N humans will
have existed. In the absence of such an estimate, the DA leaves room for alternative
interpretations. If the world population levels out at say 12 billion then disaster is
likely to put an end to our species fairly soon (within 1400 years with 75% probability).
If population figures rise higher, the prognosis is even worse. But if population
decreases drastically, then the delta t argument could be compatible with survival
for many millions of years. However, such a population collapse could perhaps itself be
called a "doomsday".

The probability of space colonization looks abysmal in the light of the Gotts
version of the DA. Reasoning via the delta t argument, Gott concludes that the
probability that we will colonize the galaxy is of order P  10-9, since
if we did manage such a feat we would expect there to be at least a billion times more
humans in the future than have been born to date.

The doomsday argument as presented by John Leslie

Leslies presentation of the DA differs in several respects from
Gotts. On a stylistic level, Leslie makes less use of mathematics than does Gott.
Leslies writing is informal and his arguments often take the form of analogies or
thought experiment. Leslie is, however, much more explicit about the philosophical
underpinnings. He places the argument in a Bayesian framework and devotes considerable
attention to the empirical considerations that determine what the priors are. One
important feature of Leslies approach is his doctrine of how the DA would be
affected if the world happens to be radically indeterministic.

The doomsday argument à la Leslie

Leslies version runs as follows:

One might at first expect the human race to survive, no doubt in evolutionary much
modified form, for millions or even billions of years, perhaps just on Earth but, more
plausibly, in huge colonies scattered through the galaxy and maybe even through many
galaxies. Contemplating the entire history of the race  future as well as past
history  I should in that case see myself as a very unusually early human. I
might well be among the first 0.00001 per cent to live their lives. But what if the race
is instead about to die out? I am then a fairly typical human. Recent population growth
has been so rapid that, of all human lives lived to far, anything up to about 30 per cent
... are lives which are being lived at this very moment. Now, whenever lacking evidence
to the contrary one should prefer to think of ones own position as fairly typical
rather than highly untypical. To promote the reasonable aim of making it quite
ordinary that I exist where I do in human history, let me therefore assume that the human
race will rapidly die out. ([1990], pp. 65f; emphasis in the original.)

Leslie emphasizes the point that the DA does not show that Doom will strike
soon. It only argues for a probability shift. If we started out being extremely
certain that the humans species will survive for a long time, we might still be fairly
certain after having taken the DA into account  though less certain than before.
Also, it is possible for us to improve our prospects. Leslie hopes that if the DA
convinces us that the risks are greater than was previously thought then we should become
more willing to take steps to diminish the dangers  perhaps through protecting the
ozone layer, pushing for nuclear disarmament, setting up a meteor early warning system, or
being careful with future very-high-energy particle physics experiments which could
possibly upset a metastable vacuum and destroy the world. So Leslie does not see the DA as
a reason for despair, but rather as a call for greater caution and concern about potential
species-annihilating disasters. (People who think that too little is done today to
safeguard against the possible extinction of our species might agree with this
recommendation even if they dont themselves believe in the DA. They could even use
the DA as an ad hominem for people who do believe in it.)

A large part of The End of the World, Leslies monograph on the doomsday
argument, consists of an examination of various concrete potential threats to our
survival. Such an examination is necessary if we are to derive some definite prediction
from the DA, since we will only get a realistic posterior probability distribution if we
put in a realistic prior. It is convenient, however, not to regard these empirical
considerations as part of the DA as such. It is more reasonable to define the DA to be
just the part of the reasoning that argues that you should not, ceteris paribus, expect to
be an untypical human observer and goes from there to argue that the risk of human
extinction has been systematically underestimated. This is where anthropic deliberation
comes in, and the philosophical problems associated with this line of reasoning can
profitably be separated out from the empirical question of how likely, say, an all-out
nuclear war is to wipe out species. This reading is generally consistent with what various
authors have written on the topic.

This is not to in any way to downplay the importance of delving into the empirical
evaluation of the risk factors. What makes the DA so important is that it gives a strong
prediction about an issue that we care a lot about. Abstracting from the empirical
content, we are left with a mere philosophical puzzle. So what we want to do is, once we
have solved the philosophical puzzle in its pure form, we want to connect it back to the
empirical information we have and see what concrete implications there might be for human
policy making and rational expectations about our future. Since this is our goal (or at
least my goal), empirical considerations will be included in the discussion; I will,
however, occasionally set them to one side in order to focus on the underlying logic of
the reasoning.

Often, Leslies arguments take the shape of a thought experiment in which it is
supposed to be intuitively clear that the rational judgement to make is in accordance with
what is required for the DA to work. Many of his thought experiments are variations on the
following theme:

Imagine an experiment planned as follows. At some point in time, three humans would
each be given an emerald. Several centuries afterwards, when a completely different set of
humans was alive, five thousand humans would each be given an emerald. Imagine next that
you have yourself been given an emerald in the experiment. You have no knowledge, however,
of whether your century is the earlier century in which just three people were to be in
this situation, or in the later century in which five thousand were to be in it. ...

Suppose you in fact betted that you lived [in the earlier century]. If every
emerald-getter in the experiment betted in this way, there would be five thousand losers
and only three winners. The sensible bet, therefore, is that yours is instead the later
century of the two. (p. 20)

Leslie introduces this example to refute the objection that the DA fails because future
humans arent yet alive so one couldnt possibly have found oneself being one of
them. It can also be used to counter several other simple objections, such as:

We cannot move around in time as we do in space, so temporal position cant be
treated as analogous to spatial position. (p.214)

(Some people could be tempted to make that objection after having read only about the
analogous spatial form of the thought experiments where the batches exist at the same
time.)

People like us are to be found only nowadays. Our characteristics force us to occupy
this era and not another. (p. 221)

(This latter objection was actually advanced in a recent Mind-paper by Korb
& Oliver, as we shall see in a later section.)

Note that what Leslies example shows that following the recommended line of
reasoning will increase the fraction of winners to losers. At this point, the argument
ends, maybe because Leslie deems that a point has been reached where any reasonable
objector would roll over and realize he was wrong.

But maybe there are other ways of making guesses that would give as good or better
result than the recommended one? Or one could perhaps object on the ground that it might
not be obvious that just because one principle maximizes the number of people who are
right, this means that it is rational for a particular individual in a particular
situation to use that principle? Leslie never attempts to go beyond analogies and give a
more rigorous formulation of the DA. It seems that at this stage it would be worthwhile to
sharpen up the debate a bit by introducing a little more rigor. To actually write down a
doomsday argument, and argue for each step, rather than just give a sketch of an argument
and then try to patch it up with analogies. This hasnt been done to date. (I will
try to it that in a later chapter.)

Leslie on the problem with the reference class

In my opinion, a major open question in observer self-selection in general and for the
DA in particular is how to define the reference class: what should count as an observer
for the purposes of the DA?

Looking backwards in time, we see a big stretch of human prehistory where it is not
clear whether our ancestors who were living then should be called "human".
Its not just that we dont know; its that the decision where to draw the
line seems to be largely arbitrary and conventional. Yet, the prediction that the DA sets
out to establish is not conventional. The odds that nuclear war will wipe out intelligent
life on Earth should not depend on how paleontologists choose to classify some old bones.

Looking in the future direction, the zone of uncertainty of what counts as an observer
is even greater. There we have to take into account the possibility that humans evolve
into posthuman life-forms. Will artificial intelligences count as observers? If so, what
kinds of these artilects will count? Should smarter, more comprehensive minds be given
more weight than less intelligent beings? What if the conventional principles that we use
to individuate minds become inapplicable due to increased bandwidth of communication that
allows minds to share memories, to copy parts of each other, to fuse, or delegate some
part of their normally conscious functions to separate and autonomous agents? If the DA is
to give us any concrete information about the future, we want to have at least the outline
of an answer to these questions.

The very difficulty of thinking of a way to settle these questions may even encourage
us to doubt the validity of the DA itself, not just to be uncertain about exactly what it
would show if it were right. For these are questions that it could seem that there ought
to be an objectively right answer to if the DA is right. It would be strange if there was
no fact of the matter about such crucial parameters as whether more intelligent minds
should be given more weight (i.e. a higher sampling density in the set of all observers)
than less intelligent minds. If there is no fact of the matter about such things then one
would have an additional reason for suspecting that the whole DA, and perhaps many other
forms of anthropic reasoning as well, were built of air and rested on some sort of
confusion. (This suspicion could be overridden if a very close examination showed that
there was nothing wrong with the DA after all; but it would still encourage some degree of
lingering metalevel doubt.)

So how does Leslie answer the question of how the reference class should be determined?

As a first remark, Leslie suggests that "perhaps nothing too much hangs on
it." (p. 257):

[The DA] can give us an important warning even if we confine our attention to the human
races chances of surviving for the next few centuries. All the signs are that these
centuries would be heavily populated if the race met with no disaster, and they are
centuries during which there would presumably be little chance of transferring human
thought-processes to machines in a way which would encourage people to call the machines
human. (p. 258)

This clearly wont do as a reply. First, the premise that there is little chance
of creating machines with human-level and human-like thought processes within the next few
centuries is something that many of those who have thought seriously about these things
would dispute. Many thinkers in this field think that these developments will happen well
within the first half of the next century (Moravec [1998a, 1998b, 1988], Drexler [1985],
Minsky [1994], Bostrom [1997a, 1997b]).

Second, the comment does nothing to soothe the suspicion that the difficulty of
determining an appropriate reference class might be symptomatic of an underlying more
fundamental difficulty with the DA itself.

Leslie does proceed, however, to offer a positive proposal for how to settle the
question of which reference class to choose.

The first part of this proposal is best understood by expanding the urn analogy that we
used to introduce the DA. Suppose that the balls in the urns came in different colors. And
suppose your task was to guess how many red balls there are in the urn in front of you.
Now, red is clearly a vague concept  what shades of pink or purple count
as red? This vagueness could be seen as corresponding to the vagueness about what to
classify as an observer for the purposes of the DA. So, if some vagueness like this is
present in the urn example, does that mean that the Bayesian induction used in the
original example can no longer be made to work at all? Clearly not.

The right response in this case is that you have a choice of how you want to define the
reference class. Your choice depends on what hypothesis you are interested in testing.
Suppose that what you are interested in finding out is how many balls there are in the urn
of the color light-pink-to-dark-purple. Then all you have to do is to classify the random
sample you select as being either light-pink-to-dark-purple or not
light-pink-to-dark-purple. Once you have made this classification, the Bayesian
calculation proceeds exactly as before. If instead you are interested in knowing how many
light-pink-to-light-red balls there are, then you classify the sample according to whether
it has that property; and then you proceed as before. The Bayesian apparatus is
perfectly neutral as to how you define the hypotheses. There is not a right or wrong way
here, just different questions you might be interested in asking.

Applying this reasoning to the DA, Leslie writes:

The moral could seem to be that ones reference class might be made more or less
what one liked for doomsday argument purposes. What if one wanted to count our
much-modified descendants, perhaps with three arms or with godlike intelligence, as
genuinely human? There would be nothing wrong with this. Yet if we were
instead interested in the future only of two-armed humans, or of humans with intelligence
much like that of humans today, then there would be nothing wrong in refusing to count any
others. (p. 260)

This suggests that if we are interested in the survival-prospects of just a special
kind of observers, we are entitled to apply the DA to this subset of the reference class.
Suppose you are black and you want to know how many black people there will have been.
Answer: Count the number of black people that have existed before you, and use the
doomsday-style calculation to update your prior conditional (given by empirical
considerations) to take account of the fact that this random sample from the set of all
blacks  you  turned out to live when just so many blacks have yet
lived.

How far can we push this mode of reasoning though, before we end up in absurdity? What
if I want to know how many people-born-on-the-tenth-of-March-in-1973-or-later there will
have been and decide to use as reference class the set of all people
born-on-the-tenth-of-March-in-1973-or-later. My temporal position among the people in this
set is extraordinarily early and will quickly become even more extraordinarily early if
humans continue to be born for much longer. Should I therefore conclude that the
population of people-born-on-the-tenth-of-March-in-1973-or-later will almost certainly go
extinct within a few years? That would obviously be absurd!

How can the doomsdayer avoid this absurd conclusion? According to Leslie, by adjusting
the prior probabilities in a suitable way:

No inappropriately frightening doomsday argument will result from narrowing your
reference class ... provided you adjust your prior probabilities accordingly. Imagine that
you'd been born knowing all about Bayesian calculations and about human history. The prior
probability of the human race ending in the very week you were born ought presumably to
have struck you as extremely tiny. And that's quite enough to allow us to say the
following: that although, if the human race had been going to last for another century,
people born in the week in question would have been exceptionally early in the class of
those-born-either-in-that-week-or-in-the-following-century, this would have been a poor
reason for you to expect the race to end in that week, instead of lasting for another
century. (p. 262)

I will criticize this solution in a later chapter and suggest another solution that I
think does the trick.

The possibility of choosing too a narrow reference class is only half of the problem
with the reference class. It is also possible to choose too a wide reference class, so we
need to know how much we can include. What about pre-historic humans? Neanderthals? Our
common ancestors with modern apes? Do these guys count as observers? Where do we draw the
line? (And again, the gray-area might be much more extensive in the future direction.)
Writes Leslie,

Widening of the reference classes can easily be taken too far. For example, we ought to
think twice before accepting any widening which counted as observers even
primitive forms of animal life. These might not be conscious at all. Furthermore it could
be held that full consciousness involves introspective ability of a kind which chimpanzees
havent yet acquired. (pp. 260-1)

This is about all that Leslie says about the problem of excessive widening of the
reference class. It appears that he thinks that having "full consciousness" is a
necessary requirement for being counted as an "observer". This condition is too
vague to be very useful.

Once we have settled on an appropriately justified reference class we have still not
reached the end of our troubles. We will also need to select and justify some particular
sampling density over the chosen reference class. This is a problem that Leslie does not
address. He implicitly assumes a uniform sampling density, i.e. that your prior
probability that you are observer X should be the same for all X in the reference class.
But this could be disputed. Perhaps clarity of mind, long life span, or time spent
thinking about the DA should result in an observer being given more weight, i.e. having a
higher sampling density in the reference class? Or maybe not, but it is by no means
obvious that the uniform distribution is always the right one. And before we specify the
sampling density we cant derive any prediction from the DA. We will come back to
this issue in a later chapter. For now its enough to note that there is a
considerable gap at this point in Leslies reasoning.

Leslie on the effect on the doomsday argument of physical indeterminism

One prominent feature of Leslies exposition of the DA is that throughout he keeps
stressing that if the world is indeterministic, as quantum physics might lead us to
believe, then the DA is seriously weakened though not completely obliterated. We shall
return to Leslies reasoning about this when we discuss the shooting room paradox.

Attempted refutations of the doomsday argument

There have been many attempted refutations of the DA, yet no one refutation
seems to have convinced many people. Most of the purported refutations can easily be seen
to be wrong, but there are a few that are more serious. I survey below those objections
that I deem to be most serious. These include all objections that have actually been
raised (as opposed to merely reported) in academic publications, with the exception of the
five objections by Korb and Oliver. (I will try to show in a later chapter that those four
objections can easily be seen to be wrong.)

The self-indication assumption

The idea behind this objection is that the probability shift in favor of earlier doom
that the DA leads us to make is offset by another probability shift that likewise has been
overlooked. This other probability shift is in the direction of a greater
probability for the hypothesis that there will have been many humans. According to this
objection, the more humans there will ever have existed, the more "slots" would
there be that you could have been "born into". Your existence is more probable
if there are many humans (or observers) than if there are few. Since you do in fact exist,
the Bayesian rule has to be applied and the posterior probability of the hypothesis
according to which many people exist must be increased.

The neat thing is that these two probability shifts cancel each other precisely,
as first noted by Dieks [1992] and shown by Kopf et al. [1994].

The principle that this objection depends on can be dubbed the self-indication
assumption:

(SIA) The fact that you are an observer gives you some reason to believe that the world
contains many observers.

Whether the objection succeeds depends on how strong reasons can be given for accepting
or rejecting this assumption. Leslie argues ([1996], pp. 224-8) that adopting SIA, we have
to conclude that the probability that the world contains infinitely many observers is one,
and that this is an unacceptable consequence. There are also other considerations that
make SIA hard to accept. We will discuss SIA further in a later chapter.

Andrei Lindes suggestion

Andrei Linde first suggested an interesting variant of the objection based on the
self-indication assumption. He thinks the universe is such that it is technologically
feasible for the human species to continue for infinitely long (see also Tipler &
Barrow [1986] and Tipler [1994]). If that is right then no matter when you were born you
would still be "infinitely early". Finding yourself alive in the late twentieth
century would be no more improbable, conditional on this hypothesis, than finding yourself
alive in, say, year 34,898, 836 AD. The DA would therefore not yield any probability
shift, although it would still be formally valid.

Leslies reply is that if we arent initially certain that the universe
contains infinitely many observers then the fact that on Lindes theory we would in
some sense be "infinitely early" gives us "superbly strong probabilistic
grounds for rejecting the theory" (Leslie [1996], p. 264). Note that it looks as if
this reply is implausible in a way symmetric to the alleged implausibility of the
SIA-objection. The SIA implied that the probability of there being infinitely many
observers is one; which seemed wrong. Now Leslies reply to Linde implies that the
probability of there being infinitely many observers is zero, which seems equally wrong.

Infinities create a lot of problems in probability theory and decision theory in many
contexts. Just think of Pascals wager, the St Petersburg paradox etc. As we shall
see, part of the puzzlement in the so-called Shooting Room paradox also derives from the
presence of infinite possibilities.

No meaningful objective probabilities

An objection that I have heard several people advance is that the DA requires the
existence of determinate probabilities where none exist. This objection may be phrased in
different ways, but the basic sentiment is as expressed by Torbjörn Tännsjö:

Leslie may well find it very improbable that we are born exceptionally early in the
history of the species (what we speak of here are subjective probabilities), but I
dont. When he claims that no observer should at all expect to find that he or
she or it had to come into existence very exceptionally early in his, her or its
species, I fully agree. But this does not mean that he or she or it should expect
not to have come into existence very exceptionally in his, her, or its species either.
The most natural attitude to adopt here is agnosticism. What we are contemplating is a
matter of radical uncertainty, not risk. (Tännsjö [1996], p. 248)

I expect that this sort of objection should look more attractive to people who
arent Bayesians. But its true that one cant take the meaningfulness of
probability assignments for granted in these kinds of very unusual applications. A proper
presentation of the DA should contain some account of why and how the probability
assignments it postulates make sense and are the right ones.

Even for a person who is convinced that the DA is valid, there is an important problem
in determining exactly what probability assignments make sense in this context. The reason
is that there is a connection here to the problem of the reference class; or so at least I
shall argue in a later chapter.

Prima facie implausibility

One obvious objection (also in Tännsjös [1996], p. 249) against the DA is that
it leads to an intuitively surprising/implausible conclusion. This is, of course, one
reason to be somewhat reluctant to immediately accept it as valid. Most people, when they
first hear about the DA, think that it is wrong. That is a perfectly healthy reaction. We
are asked to make major changes to our worldview and we rightly demand a pretty good
justification before conceding anything to the doomsdayer.

Its also clear, however, that this defense only goes so far. Thinking would be
boring if it didnt occasionally lead us to accept conclusions that originally seemed
implausible. Some level of lingering metalevel doubt may be appropriate, but at least for
the sake of philosophical discussion the burden of proof now rests equally on those who
believe in the DA and those who dont. The issue is no longer about whether the DA
should be taken seriously. We already know that the DA is interesting enough that it would
be worth refuting if it were false.

(Is a perceived need to soften the bite of the DA part of the reason why Leslie argues
that physical indeterminism will dramatically reduce the probability shift that the DA
requires? With this modification, the DA is agreeably spicy, yet not so absurdly hot as to
be impossible to swallow.)

Interpreting the doomsday argument: alternative conclusions

The DA might be seen to set out to establish that terrestrial intelligent
life (by which I include all possible future intelligent life forms that might live
off-earth but descended from us Earth-bound humans) is likely to go extinct fairly soon.
If this is the conclusion that is aimed for, then it is not at all clear that it succeeds,
even is the basic structure of the DA is correct. The reason is that seem to be alternative
conclusions, each of which is a possible way of accommodating the DA. We dont
know what the DA really shows until we have decided which of the alternative conclusions
(or which disjunction of alternative conclusions) is the right lesson to draw from the DA.
Here are some of these possible alternative conclusions:

Swamping by other considerations

The DA can be overridden if we have sufficiently strong empirical grounds for thinking
that a doomsday wont happen. This is most clearly seen in Leslies version of
the argument. If the prior we feed into Bayes formula for the hypothesis that we
will go extinct within, say 50,000 years, is small enough, then even after taking the DA
into account we can still have a very high degree of confidence that we will not go
extinct within this period.

This is all well as far as it goes. Seen as a refutation of the DA, however, it
doesnt go very far.

First, it is not an objection against the DA, but rather a point about how the DA
should be interpreted.

Second, even with a very big prior probability of survival, the posterior will become
desperately tiny for a large range of scenarios. These scenarios include those suggested
by transhumanists and others, who think that either ourselves or our electronic successors
will go on to colonize the galaxy and beyond. If that happens then there could well be
much more than 1010 times as many observers that have existed so far. Even if
our prior estimate of the likelihood that space colonization would fail were as low as one
in a million, we would still become virtually certain that large-scale space colonization
will not happen after we take the DA into account.

The point is that even with very advantageous priors, every scenario that implies the
existence of very many observers will become refuted with virtual certainty by the DA. It
doesnt matter how good our other sources of information are (within limits); for
sufficiently long durations of the human species, the posterior probability will approach
zero. (That this is the case can be easily seen by inspecting Bayes formula.) So
even though low empirical priors can reassure us for the near-time future, it doesnt
help for the long run.

This conclusion is subject to certain qualifications that may prove absolutely
decisive. We examine them below. They are based on the fact that there seem to be other
possible interpretations of what the DA shows than the one that is typically associated
with the DA (i.e. that our species will soon go extinct). When the swamping-consideration
is combined by one or more of these other considerations, it is possible that one could
consistently (and maybe even plausibly) interpret the DA as not giving us good
grounds for thinking that doomsday will happen either in the near-future or in the
far-future.

Infinite duration of the human species

We have already mentioned Lindes suggestion that the universe will allow
the human species to continue to exist forever. If that is true then one can argue for the
position that the DA is valid but inapplicable, since everybody would be in some sense
"equally early" in an ever-lasting species.

Indeterminism

We have also mentioned Leslies doctrine that quantum-mechanical
indeterminism leads to a major weakening of the DA.

Decrease in birth rates

Another possibility, if the reference class consists of all humans or all
observers that will ever have existed, is that we will turn out to be fairly typically
positioned in the reference class, not because some disaster causes us to go extinct, but
because a decrease in population. This could in itself be calamitous if the decrease were
a result of a total collapse of civilization as a result of nuclear war or an ecological
breakdown, for example. But it is possible to imagine a scenario where population figures
are voluntarily reduced. The barriers that separate one human mind from another might
begin to erode once we start to create direct links between our brains, on the one hand,
and between our brains and computers on the other. Neuro/chip interfaces are already under
development, and it has been argued that molecular nanotechnology will in all likelihood
make it possible to upload the biological neural network (the human mind) unto an
artificial neural network perhaps running as a simulation on a computer (Drexler [1985]).
This would be done by creating a 3-d scan of the human brain to an atomic level of
resolution. Once we exist as uploads, its imaginable that high-bandwidth
communication, and the ability to change mental parameters at will, and perhaps to paste
and copy cognitive modules from one individual to another, will lead to a gradual fusion
of all minds into one.

This loophole might be blocked if the future global mind runs at an extremely high
clock speed and if as a reference class we use "observer-moments" (i.e.
time-segments of observers) rather than observers. For then, even if there were just one
mind, it would quickly accumulate many observer-moments. I shall return in a later chapter
to the question of whether the reference class should consist of observers or
observer-moments.

Metamorphosis

However, there is still the ambiguity of what counts as an
observer/observer-moment. When will the mental life of our successors have changed so much
that they dont qualify as observers for the purposes of the DA? We cant
conclusively answer that question until we have settled the problem of the reference
class.

It is by no means implausible that human descendants will evolve or technologically
metamorphose into something very different from our current human form. For example,
Alexander Chislenko (in a commentary [1996] on a forthcoming book by Hans Moravec [1998])
envisions that biological intelligences will become obsolete and that the society of the
future will be a kind of functional soup populated by "infomorphs", distributed
information-processing entities existing on vast computer networks. The infomorphs would
be of varying degrees of complexity and durability; they might be able to buy and sell
knowledge and share many functions with each other. The human concept of personal identity
might not be at all useful in such a world. It would be extremely hard to determine what
should go into the reference class. It would not be clear what complexes should count as
observers or how to individuate these observers.

One would be tempted to say that the DA is not applicable to these kinds of entities,
that they should not be counted in the reference class. If thats so, then the DA
doesnt have any effect on hypotheses according to which our society will soon be
replaced by such functional soup. (Hypotheses according to which this metamorphosis is
further in the future will still be partially affected by the DA; the more so the more
humans or other clear-cut observers they imply will exist before the metamorphosis.)

One can perhaps imagine much less radical transformations than the one suggested by
Chislenko that would still be sufficient to turn us into something that falls outside the
reference class and is thus immune to the DA. Exactly how much of a metamorphosis is
necessary in order to put us outside the reference class cannot be specified until we have
solved the reference class problem.

There is a dilemma facing anybody who looks for the metamorphosis-interpretation to
lift the gloom from our outlook on the future. The dilemma is that if the metamorphosis is
too small, then the beings we metamorphose in would still be in the reference class; but
if the metamorphosis is too big, the beings we metamorphose into will not be us or
even the sort of beings we care about. The infomorphs, for example, could seem too inhuman
to give us much comfort.

Yet, it is by no means obvious that the borders of the reference class coincide with
the borders of what we care about. If they dont, then there could be room for
hypotheses that could be largely unaffected by the DA and still leave room for arbitrarily
many beings, of the sorts we care about, to exist in the future. But the reference class
problem has to be solved before this issue can be settled conclusively.

Modifying the priors through by considering a larger hypothesis space

It has been argued by Dieks [1992], Korb & Oliver [1998], and Eastmond
[1997] that we can obviate the dark conclusion of the DA simply by considering a larger
hypothesis space. If we consider only the hypotheses h1, h2, ,
hn, where hi says that there will have been i observers, and we
assume a uniform distribution over the chosen hypothesis space, then we can push the
expected number of observers upward by making n larger. The same thing happen, although to
a less degree, if we use a prior distribution such as h ´ (1/(n2)) (where h is
a normalizing constant) or if we use the so-called "unbiassed" improper prior,
1/n.

Does this show that the conclusion of the DA is arbitrary? I will argue in a later
chapter that it does not. The priors are not arbitrary or conventional  they are
supposed to represent our actual empirical knowledge of the situation. I therefore think
that this last alternative conclusion is definitely flawed.

The idea of annulling the DA by introducing the self-indication assumption could also
be characterized as a proposal to change the priors from what we would naively take them
to be. But that case is different since an independent motivation was supplied there for
that particular choice of prior. It was not a matter of changing the prior for the sole
purpose of avoiding the standard DA conclusion.

The shooting room paradox

The shooting-room paradox was introduced by John Leslie (e.g. [1996], pp.
251ff), who says he developed the idea with help from David Lewis, who considers it
"a good, hard paradox".

In the shooting room experiment we are to imagine a room of infinite capacity. First a
batch of ten people are led into this room. A pair of dice is thrown in front of their
eyes. If a double six comes up they are all shot. Otherwise they leave the room safely and
a new batch, this one containing a hundred people, is thrust in. The process continues,
with each consecutive batch ten times larger than the previous one, until there is a
double six; whereupon the people in the room at that time are shot and the experiment
ends.

Suppose you have been thrust into the room. You are asked to estimate the odds of
leaving safely. One the one hand, since whether you will leave or not will be determined
by the throw of a fair pair of dice, it seems that you have a 35/36 chance of exiting
alive. On the other hand, 90% of all people who are in your situation will be shot, so it
seems you have only a 10% chance of exiting alive. That is the paradox.

The connection to the DA is obvious. Except for the fact that each consecutive batch in
the shooting room is postulated to be ten times bigger than its predecessor (which
corresponds to an indefinite exponential population growth in the case of the DA), the two
situations are structurally very similar.

Leslie on the shooting room paradox

Leslie thinks there is a radical difference for what the person in the shooting room
should believe depending on whether the random mechanism (the two dice) are deterministic
or not.

Consider first the indeterministic case. Suppose the outcome of the dice is determined
by a radically indeterministic quantum process. (And we suppose that everybody is aware of
this fact.) Then according to Leslie you should expect to get out of the room alive.
Leslie admits that "this is in a way wildly paradoxical, given that at least 90 per
cent of those who betted they would get out alive would lose their bets." (Leslie
[1996], p.252). As Bas van Fraassen remarked, an insurance agent who insured all of them
would be making a costly mistake. Despite this, Leslie maintains that the chances of not
being shot are 35/36:

It can nevertheless seem fairly plain that you personally should expect the dice not to
fall double-six if you do know for sure that they are fair, radically indeterministic
dice. For there the dice are, resting in the Devils hand; they havent yet been
thrown; and there is either no fact of the matter of how they are going to
fall, or else (see the discussion [in an earlier chapter of The End of the World]
of the irrelevance of the B-theory of Time) no fact of the matter to which you can
properly appeal. All you can say is that you have thirty-five chances out of thirty-six of
leaving the room safely. End of argument, I think. (Leslie [1996], pp. 252-3).

This result is supposed to hold under the assumption that the Devil can continue to
create new people forever, and fit them into the room, so that the process can continue
indefinitely. (Leslie thinks the result would also hold if this assumption is relaxed and
we stipulate some finite but very large maximal number of people that could enter the room
(whereafter the experiment would end); he does not press that point, however.)

Next, consider the deterministic case. For example, instead of the dice we might use
two consecutive decimal places in the expansion of of pi. These decimal places could be
chosen to be far away from what anybody has ever calculated, and we could disregard all
decimals that are not between one and six.

Here Leslies verdict is reversed: if you enter the room under these conditions,
expect to be shot  "For now theres no need for you to accept the
paradoxical conclusion which seemed forced on you in the indeterministic version of the
experiment. You cannot say that, when you arrived in the room, whether youd exit
from it safely hadnt yet been fixed by factors working deterministically." (p.
254). And "Disaster is what will come to over 90 per cent of those who will ever have
been in your situation." (p. 255).

There is a connection between Leslies views on the shooting room paradox and his
doctrine of "observer-relative chances". In the indeterministic case of the
shooting room, how can it be right that you should expect to come out alive, while at the
same time an insurance company wanting to insure all people who entered the room would be
making a costly mistake? Leslies answer is that chances are observer-relative in a
paradoxical way, so that the rational probability estimates of how the dice are likely to
fall will differ depending on weather you are the insurance company or a person in the
room.

In a later chapter I will critically examine Leslies views on the shooting room,
and I will also argue that the paradox can be resolved without any appeal to paradoxical
observer-relative chances.

Eckhardts critique

William Eckhardt [1997] argues against Leslies doctrine that the probabilities in
the deterministic shooting room are different from the ones in the indeterministic
variant. Eckhardt thinks that the probability in each case is 35/36 that you will get out
alive, conditional on finding yourself in the shooting room. In Eckhardts view,
"the shooting room is not a paradox at all; rather, it is a cogent line of reasoning
alongside an utterly spurious one, masquerading as horns of a dilemma." (p. 253).

How does this square with the fact that a booker who betted against all the people in
the room at these odds would be certain to make a profit? Since the bookies gain are
the punters loss, and if both expect to make a profit, then it would seem as if at
least one of them were mistaken about the odds. Eckhardt points out that this problem only
arises in an imaginary world where the house, if need be, can continue to raise the stakes
forever. In the real world, the house would risk running out of credit. So what we have
here is basically a pyramid scam. "It has long been known that by successively
increasing bet size in a sequence of unfavorable bets, one can theoretically obtain
winning results; [footnote omitted] this is the basis of various infamous doubling systems
in roulette and other games." (Eckhardt [1997], p. 253.)

To suppose that the odds depend on whether the random mechanism is deterministic or not
has unpalatable consequences according to Eckhardt:

If there existed a mode of statistical inference that were valid according to the
extent that determinism were true then by repeatedly testing the accuracy of this
type of statistical inference, one could gauge the correctness of determinism. Since this
conclusion is highly implausible, it is a safe bet that statistical inferences, including
those which underlie the doomsday argument, do not hinge on the truth of determinism.
(Eckhardt [1997], pp. 245-6.)

Since this objection presupposes repeatability, it doesnt immediately strike
Leslies doctrine on the shooting room with full force. At least in its original
formulation, the shooting room cannot be repeated, since once there is a double-six,
youll be shot. Leslie, however, seems to think that the same point about the
relevance of physical determinism can be made also in other contexts where the game is
repeatable. In that case it could look as if Leslie is forced into the unattractive
position of having to maintain that physicists could set up some kind of gambling
institution in order to ascertain whether the world is indeterministic or not.

I shall later argue that there is a clear sense in which unrepeatability is an
essential feature of observer self-selection.

A second objection that Eckhardt advances against Leslies view on the relevance
of determinism is as follows. We can imagine that a sequence of indeterministic dice
throws is recorded. Then we have two cases. In one shooting room, the fate of the people
that enter is determined directly by the original indeterministic process. In another
shooting room, at some later time, the fate of the people that enter this room is
determined by the transcripts of the outcome of the indeterministic process. The sequence
of dice outcomes is the same in both cases, and we can assume that the people involved
know this. Yet, according to Leslie, what you should believe depends on whether you are in
the deterministic shooting room or the indeterministic one. Thus, Leslies claims
lead to contradicting statements about what amount to the same game; i.e. Leslies
doctrine is self-refuting.

Leslie responds to this briefly in a footnote. He dismisses the objection as
question-begging: "I reject Eckhardts question-begging claim that betting games
are the same games regardless of whether the are played (a) with
indeterministic dice or else (b) with records" (Leslie [1997], pp. 435f). While this
defense might save Leslie from outright contradiction, it doesnt remove the
perceived implausibility of the consequence that he is committed to treating the two games
differently.